La Profilée - The Impossibility of Alternative Persistence Regimes: On the Exhaustion of Coherent Formulations of Persistence

Paper 122 establishes that the admissibility conditions of the persistence problem (C1-C3) are not chosen but structurally induced by the possibility of forming a persistence verdict at all. This paper addresses the final remaining question: can persistence be formulated coherently under a different admissibility regime? The result is negative. All alternative formulations of persistence fall into one of three classes: (1) Trivialization — persistence is guaranteed and the problem disappears; (2) Indeterminacy — persistence cannot be decided; (3) Equivalence — the formulation reduces to the LP structure. No fourth class exists. Therefore, the LP admissibility regime is not one possible formulation of persistence. It is the only structurally coherent formulation of the problem.